integral of x^2(1-x)^{1/2}

Lösung

\int x^{2} (1 - x)^{\frac{1}{2}} d x

- 2 (\frac{1}{7} (1 - x)^{\frac{7}{2}} - \frac{2}{5} (1 - x)^{\frac{5}{2}} + \frac{1}{3} (1 - x)^{\frac{3}{2}}) + C

Schritte zur Lösung

\int x^{2} (1 - x)^{\frac{1}{2}} d x

Wende U-Substitution an

= \int - 2 (- u^{2} + 1)^{2} u^{2} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 2 \cdot \int (- u^{2} + 1)^{2} u^{2} d u

Multipliziere aus

(- u^{2} + 1)^{2} u^{2} : u^{6} - 2 u^{4} + u^{2}

= - 2 \cdot \int u^{6} - 2 u^{4} + u^{2} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - 2 (\int u^{6} d u - \int 2 u^{4} d u + \int u^{2} d u)

\int u^{6} d u = \frac{u ^{7}}{7}

\int 2 u^{4} d u = \frac{2 u ^{5}}{5}

\int u^{2} d u = \frac{u ^{3}}{3}

= - 2 (\frac{u ^{7}}{7} - \frac{2 u ^{5}}{5} + \frac{u ^{3}}{3})

Setze in

u = (1 - x)^{\frac{1}{2}}

ein

= - 2 \frac{( ( 1 - x ) ^{\frac{1}{2}} ) ^{7}}{7} - \frac{2 ( ( 1 - x ) ^{\frac{1}{2}} ) ^{5}}{5} + \frac{( ( 1 - x ) ^{\frac{1}{2}} ) ^{3}}{3}

Vereinfache

- 2 \frac{( ( 1 - x ) ^{\frac{1}{2}} ) ^{7}}{7} - \frac{2 ( ( 1 - x ) ^{\frac{1}{2}} ) ^{5}}{5} + \frac{( ( 1 - x ) ^{\frac{1}{2}} ) ^{3}}{3} : - 2 (\frac{1}{7} (1 - x)^{\frac{7}{2}} - \frac{2}{5} (1 - x)^{\frac{5}{2}} + \frac{1}{3} (1 - x)^{\frac{3}{2}})

= - 2 (\frac{1}{7} (1 - x)^{\frac{7}{2}} - \frac{2}{5} (1 - x)^{\frac{5}{2}} + \frac{1}{3} (1 - x)^{\frac{3}{2}})

Füge eine Konstante zur Lösung hinzu

= - 2 (\frac{1}{7} (1 - x)^{\frac{7}{2}} - \frac{2}{5} (1 - x)^{\frac{5}{2}} + \frac{1}{3} (1 - x)^{\frac{3}{2}}) + C

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